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Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height component. A point in space is expressed as

• \((r, \theta, z)\), where:
• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle relative to the positive x-axis.
• \(z\) is the height above the xy-plane.

This system is particularly useful when dealing with objects that have a cylindrical shape, as it simplifies the integral setup by aligning with the object's natural geometry.
In the current exercise, we describe a solid cylinder using cylindrical coordinates with limits of \(0 \leq r \leq 4\), \(0 \leq z \leq 10\), and \(0 \leq \theta \leq 2\pi\). Such a setup effectively captures the entire geometry of the cylinder, making it convenient for performing integration.

A density function in the context of mass calculations represents how mass is distributed within a solid object. It can vary depending on all the spatial variables involved.

• In general, the density \(\rho(r, \theta, z)\) gives the mass per unit volume at any point in the object.

For the exercise at hand, the density function is given by \(\rho(r, \theta, z) = 1 + \frac{z}{2}\), suggesting that the density increases linearly with height \(z\). As such, areas higher above the base are denser than those closer to it.
Understanding the function is crucial because it dictates how the mass is distributed along the cylinder's height, hence affecting the total mass outcome once integrated.

Mass calculation involves evaluating the sum of tiny masses over the solid's volume, which is performed by integrating the density function over the region.

• We rely on triple integration to sum the contributions from each infinitesimally small portion of the object.

In this example, after substituting the given density function \(\rho(r, \theta, z) = 1 + \frac{z}{2}\) into the integral, the task involves evaluating \(\int \int \int_D \rho(r, \theta, z) \, r \, dz \, d\theta \, dr\).
It's crucial to integrate correctly with respect to each of the cylindrical coordinates. Start with \(z\), then \(r\), and finish with \(\theta\) to arrive at the mass. This formula essentially accumulates mass across all infinitesimal sections from \(r\), \(\theta\), and \(z\). The careful setup and computation reveal a final mass of \(120\pi\) for the cylinder.

Setting up the integral in cylindrical coordinates transforms the mass calculation into something much more manageable by aligning the integration with the shape of the object.
The integration process involves:

• Defining your limits for \(r\), \(\theta\), and \(z\) based on the geometry of the object.
• Including the factor of \(r\) in the integral to account for the radial distance in cylindrical coordinates.

For the cylinder, these limits become \(0 \leq r \leq 4\), \(0 \leq z \leq 10\), and \(0 \leq \theta \leq 2\pi\), seamlessly mapping the volume to be integrated.
The triple integral form \(\int_0^{2\pi} \int_0 ^4 \int_0^{10} \rho(r,\theta,z) r \, dz \, d\theta \, dr\)ensures that you are accounting for all mass elements throughout the volume of the cylinder. Understanding and setting these correctly is essential to solving the problem and finding the object's total mass.